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Symbolic integration: towards practical algorithms. (English) Zbl 0724.12006

Computer algebra and differential equations, Colloq., Comput. Math. Appl., 59-85 (1988).
This article is an excellent survey of the problem of integration in finite terms, i.e. the problem of deciding in a finite number of steps whether an elementary function has an elementary indefinite integral and to compute it explicitly if it exists. Starting with Ch. Hermite’s rational algorithm [Sur l’intégration des fonctions rationelles, Ann. Sci. Éc. Norm. Supér. (2) 1, 215–218 (1872; JFM 04.0125.05)], the author reviews – without proofs – the original Risch algorithm [R. H. Risch, Trans. Am. Math. Soc. 139, 167–189 (1969; Zbl 0184.06702); Bull Am. Math. Soc. 76, 605–608 (1970; Zbl 0196.06801)] and the later improvements found by M. Rothstein [Proc. 1977 MACSYMA Users’ Conference, 263–274], J. H. Davenport [On the integration of algebraic functions, Lecture Notes in Computer Science. 102. Berlin etc.: Springer (1981; Zbl 0471.14009)], B. Trager [Ph. D. Thesis, Massachusetts Institute of Technology (1984)] and by himself and describes clearly how Hermite’s technique can be effectively applied to rational, algebraic, transcendental and finally to arbitrary elementary functions.
For a detailed description of the author’s algorithm – with complete proofs – see his paper [J. Symb. Comput. 9, 117–173 (1990; Zbl 0718.12006)].
For the entire collection see Zbl 0702.00010.

MSC:

12H05 Differential algebra
68W30 Symbolic computation and algebraic computation
14H05 Algebraic functions and function fields in algebraic geometry

Software:

MACSYMA